Advertisement

Weakly Consistent Regularisation Methods for Ill-Posed Problems

  • Erik Burman
  • Lauri Oksanen
Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 15)

Abstract

This Chapter takes its origin in the lecture notes for a 9 h course at the Institut Henri Poincaré in September 2016. The course was divided in three parts. In the first part, which is not included herein, the aim was to first recall some basic aspects of stabilised finite element methods for convection-diffusion problems. We focus entirely on the second and third parts which were dedicated to ill-posed problems and their approximation using stabilised finite element methods. First we introduce the concept of conditional stability. Then we consider the elliptic Cauchy-problem and a data assimilation problem in a unified setting and show how stabilised finite element methods may be used to derive error estimates that are consistent with the stability properties of the problem and the approximation properties of the finite element space. Finally, we extend the result to a data assimilation problem subject to the heat equation.

References

  1. 1.
    Alessandrini, G., Rondi, L., Rosset, E., Vessella, S.: The stability for the Cauchy problem for elliptic equations. Inverse Prob. 25(12), 123004, 47 (2009). http://dx.doi.org/10.1088/0266-5611/25/12/123004 MathSciNetCrossRefGoogle Scholar
  2. 2.
    Andrieux, S., Baranger, T.N., Ben Abda, A.: Solving Cauchy problems by minimizing an energy-like functional. Inverse Prob. 22(1), 115–133 (2006). http://dx.doi.org/10.1088/0266-5611/22/1/007 MathSciNetCrossRefGoogle Scholar
  3. 3.
    Azaï ez, M., Ben Belgacem, F., El Fekih, H.: On Cauchy’s problem. II. Completion, regularization and approximation. Inverse Prob. 22(4), 1307–1336 (2006). http://dx.doi.org/10.1088/0266-5611/22/4/012
  4. 4.
    Badra, M., Caubet, F., Dardé, J.: Stability estimates for Navier-Stokes equations and application to inverse problems. Discrete Contin. Dyn. Syst. Ser. B 21(8), 2379–2407 (2016). http://dx.doi.org/10.3934/dcdsb.2016052MathSciNetCrossRefGoogle Scholar
  5. 5.
    Baumeister, J.: Stable Solution of Inverse Problems. Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn, Braunschweig (1987). http://dx.doi.org/10.1007/978-3-322-83967-1 Google Scholar
  6. 6.
    Ben Belgacem, F., Du, D.T., Jelassi, F.: Extended-domain-Lavrentiev’s regularization for the Cauchy problem. Inverse Prob. 27(4), 045005 (2011). http://dx.doi.org/10.1088/0266-5611/27/4/045005 MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bourgeois, L.: A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace’s equation. Inverse Prob. 21(3), 1087–1104 (2005). http://dx.doi.org/10.1088/0266-5611/21/3/018 MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bourgeois, L., Dardé, J.: The “exterior approach” to solve the inverse obstacle problem for the Stokes system. Inverse Probl. Imaging 8(1), 23–51 (2014). http://dx.doi.org/10.3934/ipi.2014.8.23MathSciNetCrossRefGoogle Scholar
  9. 9.
    Brenner, S.C.: Poincaré-Friedrichs inequalities for piecewise H 1 functions. SIAM J. Numer. Anal. 41(1), 306–324 (2003). http://dx.doi.org/10.1137/S0036142902401311 MathSciNetCrossRefGoogle Scholar
  10. 10.
    Brooks, A.N., Hughes, T.J.R.: Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 32(1–3), 199–259 (1982). http://dx.doi.org/10.1016/0045-7825(82)90071-8. FENOMECH ‘81, Part I (Stuttgart, 1981)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Burman, E.: Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: Elliptic equations. SIAM J. Sci. Comput. 35(6), A2752–A2780 (2013). http://dx.doi.org/10.1137/130916862 zbMATHGoogle Scholar
  12. 12.
    Burman, E.: Error estimates for stabilized finite element methods applied to ill-posed problems. C. R. Math. Acad. Sci. Paris 352(7–8), 655–659 (2014). http://dx.doi.org/10.1016/j.crma.2014.06.008 MathSciNetCrossRefGoogle Scholar
  13. 13.
    Burman, E.: Stabilised finite element methods for ill-posed problems with conditional stability. In: Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol. 114, pp. 93–127. Springer, Cham (2016)Google Scholar
  14. 14.
    Burman, E.: The elliptic Cauchy problem revisited: control of boundary data in natural norms. C. R. Math. 355, 479–484 (2017). http://dx.doi.org/10.1016/j.crma.2017.02.014 MathSciNetCrossRefGoogle Scholar
  15. 15.
    Burman, E., Hansbo, P.: Edge stabilization for Galerkin approximations of convection-diffusion-reaction problems. Comput. Methods Appl. Mech. Eng. 193(15–16), 1437–1453 (2004). http://dx.doi.org/10.1016/j.cma.2003.12.032 MathSciNetCrossRefGoogle Scholar
  16. 16.
    Burman, E., Hansbo, P.: Stabilized nonconforming finite element methods for data assimilation in incompressible flows. Math. Comp. 87(311), 1029–1050 (2018).MathSciNetCrossRefGoogle Scholar
  17. 17.
    Burman, E., Oksanen, L.: Data assimilation for the heat equation using stabilized finite element methods. Numer. Math. 139(3), 505–528 (2018).MathSciNetCrossRefGoogle Scholar
  18. 18.
    Burman, E., Oksanen, L., Ish-Horowicz, J.: Fully discrete finite element data assimilation method for the heat equation. ESAIM: Math. Model. Numer. Anal. (2018, in press). https://doi.org/10.1051/m2an/2018030
  19. 19.
    Burman, E., Hansbo, P., Larson, M.: Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization. Inverse Problems 34(3), (2018).MathSciNetCrossRefGoogle Scholar
  20. 20.
    Cheng, J., Yamamoto, M.: One new strategy for a priori choice of regularizing parameters in Tikhonov’s regularization. Inverse Prob. 16(4), L31 (2000). http://stacks.iop.org/0266-5611/16/i=4/a=101 MathSciNetCrossRefGoogle Scholar
  21. 21.
    Dardé, J., Hannukainen, A., Hyvönen, N.: An H div-based mixed quasi-reversibility method for solving elliptic Cauchy problems. SIAM J. Numer. Anal. 51(4), 2123–2148 (2013). http://dx.doi.org/10.1137/120895123 MathSciNetCrossRefGoogle Scholar
  22. 22.
    Di Pietro, D.A., Ern, A.: Mathematical aspects of discontinuous Galerkin methods. In: Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 69. Springer, Heidelberg (2012). http://dx.doi.org/10.1007/978-3-642-22980-0
  23. 23.
    Èmanuilov, O.Y.: Controllability of parabolic equations. Math. Sb. 186(6), 109–132 (1995). http://dx.doi.org/10.1070/SM1995v186n06ABEH000047 MathSciNetCrossRefGoogle Scholar
  24. 24.
    Engl, H.W., Hanke, M., Neubauer, A.: Regularization of inverse problems. In: Mathematics and Its Applications, vol. 375. Kluwer Academic Publishers Group, Dordrecht (1996). http://dx.doi.org/10.1007/978-94-009-1740-8
  25. 25.
    Ern, A., Guermond, J.L.: Theory and practice of finite elements. In: Applied Mathematical Sciences, vol. 159. Springer, New York (2004). http://dx.doi.org/10.1007/978-1-4757-4355-5
  26. 26.
    Guermond, J.L.: Stabilization of Galerkin approximations of transport equations by subgrid modeling. M2AN Math. Model. Numer. Anal. 33(6), 1293–1316 (1999). http://dx.doi.org/10.1051/m2an:1999145 MathSciNetCrossRefGoogle Scholar
  27. 27.
    Johnson, C., Nävert, U., Pitkäranta, J.: Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Eng. 45(1–3), 285–312 (1984). http://dx.doi.org/10.1016/0045-7825(84)90158-0 MathSciNetCrossRefGoogle Scholar
  28. 28.
    Kabanikhin, S.I.: Definitions and examples of inverse and ill-posed problems. J. Inverse Ill-Posed Probl. 16(4), 317–357 (2008). http://dx.doi.org/10.1515/JIIP.2008.019MathSciNetCrossRefGoogle Scholar
  29. 29.
    Klibanov, M.V.: Carleman estimates for the regularization of ill-posed Cauchy problems. Appl. Numer. Math. 94, 46–74 (2015). http://dx.doi.org/10.1016/j.apnum.2015.02.003 MathSciNetCrossRefGoogle Scholar
  30. 30.
    Kohn, R.V., Vogelius, M.: Relaxation of a variational method for impedance computed tomography. Commun. Pure Appl. Math. 40(6), 745–777 (1987). http://dx.doi.org/10.1002/cpa.3160400605MathSciNetCrossRefGoogle Scholar
  31. 31.
    Lattès, R., Lions, J.L.: Méthode de quasi-réversibilité et applications. Travaux et Recherches Mathématiques, No. 15. Dunod, Paris (1967)Google Scholar
  32. 32.
    Puel, J.P.: A nonstandard approach to a data assimilation problem and Tychonov regularization revisited. SIAM J. Control Optim. 48(2), 1089–1111 (2009). http://dx.doi.org/10.1137/060670961 MathSciNetCrossRefGoogle Scholar
  33. 33.
    Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990). http://dx.doi.org/10.2307/2008497 MathSciNetCrossRefGoogle Scholar
  34. 34.
    Tikhonov, A.N., Arsenin, V.Y.: Solutions of Ill-Posed Problems. V. H. Winston & Sons, Washington; Wiley, New York (1977)Google Scholar
  35. 35.
    Yamamoto, M.: Carleman estimates for parabolic equations and applications. Inverse Prob. 25(12), 123013 (2009). http://dx.doi.org/10.1088/0266-5611/25/12/123013 MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK

Personalised recommendations